Mally's Deontic Logic

In 1926, Mally presented the first formal system of deontic logic.
His system had several consequences which Mally regarded as surprising
but defensible. It also had a consequence (“A is obligatory if and only
if A is the case”) which Menger (1939) and almost all later deontic
logicians have regarded as unacceptable. We will not only describe
Mally's system but also discuss how it may be repaired.

In 1926, the Austrian philosopher Ernst Mally (1879-1944) proposed
the first formal system of deontic logic. In the book in which he
presented this system, The Basic Laws of Ought: Elements of the
Logic of Willing, Mally gave the following motivation for his
enterprise:

In 1919, everybody was using the word self-determination. I
wanted to obtain a clear understanding of this word. But then, of
course, I immediately stumbled on the difficulties and obscurities
surrounding the concept of Ought, and the problem changed. The concept
of Ought is the basic concept of the whole of ethics. It can only serve
as a usable foundation for ethics when it is captured in a system of
axioms. In the following I will present such an axiomatic
system.[1]

As Mally's words indicate, he was not primarily interested in
deontic logic for its own sake: he mainly wanted to lay the foundation
of “an exact system of pure ethics” (eine exakte reine Ethik).
More than half of his book is devoted to the development of this exact
system of ethics. In the following, we will, however, concentrate on
the formal part of his book, both because it is its “hard core” and
because it is the part that has attracted the most interest.

Mally based his formal system on the classical propositional
calculus as formulated in Whitehead's and Russell's Principia
Mathematica (vol. 1, 1910).

The non-deontic part of Mally's system had the following vocabulary:
the sentential letters A, B, C, P and Q (these symbols refer to states
of affairs), the sentential variables M and N, the sentential
constants V (the Verum, Truth) and Λ
(the Falsum, Falsity), the propositional quantifiers ∃
and ∀, and the connectives ¬, &, ∨, → and
↔. Λ is defined by Λ = ¬V.

The deontic part of Mally's vocabulary consisted of the unary
connective !, the binary connectives f and ∞, and the sentential
constants U and ∩.

Mally read !A as “A ought to be the case” (A soll sein) or
as “let A be the case” (es sei A).

Mally did not only read !A as “it ought to be the case that A.”
Because a person's willing that a given state of affairs A be the case
is often expressed by sentences of the form “A ought to be the case”
(for example, someone might say “it ought to be the case that I am rich
and famous” to indicate that she wants to be rich and famous), he also
read !A as “A is desirable” or “I want it to be the case that A.” As a
result, his formal system was as much a theory about Wollen
(willing) as a theory about Sollen (ought to be the case).
This explains the subtitle of his book. In modern deontic logic, the
basic deontic connective O is seldom read in this way.

We have just described one respect in which Mally's deontic logic
was different from more modern proposals. There are two other
conspicuous differences:

Mally was only interested in the deontic status of states of
affairs; he paid no special attention to the deontic status of actions.
Thus, his Deontik was a theory about Seinsollen (what
ought to be the case) rather than Tunsollen (what ought to be
done). Modern authors often regard the concept of Tunsollen as
fundamental.

In modern deontic logic, the notions of prohibition F, permission P
and waiver W are usually defined in terms of obligation O: FA =
O¬A, PA = ¬FA, WA = ¬OA. Such definitions are not to be
found in Mally's book.

Axiom IV is the only axiom or theorem mentioned by Mally in
which U occurs as a bound variable: in Axiom V and in theorems
(15)-(17), (20)-(21), (23), (23′) and (27)-(35) (to be displayed
below), U is either a constant or a free variable. One should treat it
in the same way in the formalization of (iv).

Mally could hardly have objected to this version of Axiom IV
because it is equivalent with his theorem (23′), i.e., V f U, in
virtue of Def. f. In the following “Axiom IV” will always
refer to our version of Axiom IV rather than Mally's.

Using Def. f, Axioms I-V may also be written as follows
(Mally 1926, pp. 15-19 and p. 24):

Mally called theorems (1), (2), (7), (22) and (27)-(35) “surprising”
(befremdlich) or even “paradoxical” (paradox). He
viewed (34) and (35) as the most surprising of his surprising theorems.
But Mally's reasons for calling these theorems surprising are puzzling
if not confused.

Consider, for example, theorem (1). Mally interpreted this theorem
as follows: “if A requires B, then A requires everything that is the
case” (Mally 1926, p. 20). He regarded this as a surprising claim,
and we agree. However, Mally's interpretation of (1) is not warranted.
(1) only says that if A requires B, then A requires the Verum.
The expression “if A requires B, then A requires everything that is the
case” is to be formalized as

(1′)

(A f B) → (C → (A f C))

This formula is an immediate consequence of (1) in virtue of
Axiom I. In other words, Mally should have reasoned as follows:
(1′) is surprising; but (1′) is an immediate consequence of
(1) in virtue of Axiom I; Axiom 1 is uncontroversial; so (1)
is to be regarded as surprising.

A similar pattern is to be seen in many of Mally's other remarks
about theorems which surprised him. He generally read too much into
them and confused them with some of the consequences they had in his
system:

Mally was surprised by (2) because he thought that it says that if
A requires B and B is not the case, then A requires every state of
affairs whatsoever (Mally 1926, p. 21). But (2) says no such
thing. Mally's paraphrase is a paraphrase of (A f B) → (¬B
→ (A f C)) (a consequence of (2) in virtue of Axiom I) rather
than (2).

Mally paraphrased (7) as “if anything is required, then everything
that is the case is required” (Mally 1926, p. 28), which is indeed
surprising. However, Mally's paraphrase corresponds with !A → (B
→ !B) (a consequence of (7) in virtue of Axiom I) rather than
(7).

Mally paraphrased (22) as “the facts ought to be the case” (Mally
1926, p. 24). We grant that this is a surprising claim. But the
corresponding formula in Mally's language is A → !A (a consequence
of (22) in virtue of Axiom I), not (22).

Mally read (27) as “if something which ought not to be the case is
the case, then anything whatsoever ought to be the case” (Mally 1926,
pp. 24, 33), but this is a paraphrase of !¬A → (A →
!B) (a theorem of Mally's system) rather than (27).

Mally paraphrased (33) as “what is not the case is not obligatory”
(Mally 1926, p. 25) and as “everything that is obligatory is the
case” (Mally 1926, p. 34). These assertions are indeed surprising,
but Mally's readings of (33) are not warranted. They are paraphrases of
!A → A rather than (33).

Mally made the following remark about (34) and (35):

The latter sentences, which seem to identify being obligatory with
being the case, are surely the most surprising of our “surprising
consequences.”[4]

However, (34) and (35) do not assert that being obligatory is
equivalent with being the case, for the latter statement should be
formalized as A ↔ !A. The latter formula is a theorem of Mally's
system, as will be shown in a moment, but it is not to be found in
Mally's book.

Mally regarded theorems (28)-(32) as surprising because of their
relationships with certain other surprising theorems:

(28)-(30) are instantiations of (27). But this is not sufficient to
call these theorems surprising. Mally actually viewed (28) as less
surprising than (27): one might use it to justify retaliation and
revenge (Mally 1926, p. 24).

(31) implies (28)-(30) and is therefore at least as surprising as
those theorems.

Mally viewed (32) as surprising because the surprising theorem (33)
is an immediate consequence of (32) and the apparently non-surprising
theorem (25).

Mally's list of surprising theorems seems too short: for example,
(24) is equivalent with A → !A in virtue of Def. f. But A
→ !A may be paraphrased as “the facts ought to be the case,” an
assertion which Mally regarded as surprising (Mally 1926, p. 24).
So then why didn't he call (24) surprising? Did it not surprise him
after (22)?

Even though Mally regarded many of his theorems as surprising, he
thought that he had discovered an interesting concept of “correct
willing” (richtiges Wollen) or “willing in accordance with the
facts” which should not be confused with the notions of obligation and
willing used in ordinary discourse. Mally's “exact system of pure
ethics” was mainly concerned with this concept, but we will not
describe this system because it belongs to the field of ethics rather
than deontic logic.

Mally's enterprise was received with little enthusiasm. As early as
1926, it was noted that “Mr. Mally's deductions are frequently so
amazingly obtuse and irrelevant that (despite his elaborate symbolic
apparatus) it is only necessary to state one or two of them to show
how far his discussion has strayed from its self-appointed task”
(Laird 1926, p. 395).

In 1939, Karl Menger published a devastating attack on Mally's
system. He first pointed out that A ↔ !A is a theorem of
this system. In other words, if A is the case, then A is obligatory,
and if A ought to be the case then A is indeed the case. As we have
already noted in connection with theorems (34) and (35), Mally made the
same claim in informal terms, but the formula A ↔ !A does not
occur in his book.

Menger's theorem A ↔ !A may be proven as follows (Menger's
proof was different; PC denotes the propositional
calculus).

This result seems to me to be detrimental for Mally's theory,
however. It indicates that the introduction of the sign ! is
superfluous in the sense that it may be cancelled or inserted in any
formula at any place we please. But this result (in spite of Mally's
philosophical justification) clearly contradicts not only our use of
the word “ought” but also some of Mally's own correct remarks about
this concept, e.g. the one at the beginning of his development to the
effect that p → (!q or !r) and p → !(q or r) are not
equivalent. Mally is quite right that these two propositions are not
equivalent according to the ordinary use of the word “ought.” But they
are equivalent according to his theory by virtue of the equivalence of
p and !p (Menger 1939, p. 58).

Almost all deontic logicians have accepted Menger's verdict. After
1939, Mally's deontic system has seldom been taken seriously.

Where did Mally go wrong? How could one construct a system of
deontic logic which does more justice to the notion of obligation used
in ordinary discourse? Three types of answers are possible:

Mally should not have added his deontic axioms to classical
propositional logic;

Some of his deontic principles should be modified; and

Both of the above. Menger advocated the latter view: “One of the
reasons for the failure of Mally's interesting attempt is that it was
founded on the 2-valued calculus of propositions” (Menger 1939,
p. 59).

The first two suggestions turn out to be sufficient, so the third
proposal is overkill.

In the following we will point out three facts:

First, if Mally's deontic principles are added to a
system in which the so-called paradoxes of material and strict
implication are avoided, many of the “surprising” theorems (such as
(34) and (35)) are no longer derivable and A ↔ !A
is no longer derivable either (section 8).

Second, if Mally's deontic principles are added to a
system in which the so-called law of the excluded middle is
avoided, the unacceptable consequence A ↔ !A is no longer
derivable, but almost all theorems that Mally derived himself are still derivable
(section 9).

Third, if Mally's deontic principles, e.g., Def. f and Axiom I, are slightly modified, the resulting system is almost identical with the system nowadays known as standard deontic logic (section 10).

Mally's informal postulates (i)-(iii) and (v) are conditionals or
negations of conditionals, i.e., of the form “if … then
—” or “not: if … then —.” Føllesdal and
Hilpinen (1981, pp. 5-6) have suggested that such conditionals
should not be formalized in terms of material implication and that
some sort of strict implication would be more appropriate. But this
suggestion is not altogether satisfactory, for both A → !A and
A ↔ !A are derivable in the very weak system
S0.90 plus I′ and
III′, where → is the symbol of strict
implication.[7]

In systems of strict implication the so-called paradoxes of material
implication (such as A → (B → A)) are avoided, but the
so-called paradoxes of strict implication (such as (A & ¬A)
→ B) remain. What would happen to Mally's system if both
kinds of paradoxes were avoided? This question can be answered by
adding Mally's axioms to a system in which none of the so-called
“fallacies of relevance” can be derived (see the entry on
relevance logic).

In the following, we will add Mally's axioms to the prominent
relevance logic R. The result is
better than in the case of strict implication: most of the theorems
which Mally regarded as surprising are no longer derivable, and
Menger's theorem A ↔ !A is not derivable either. But many
“plausible” theorems can still be derived.

Relevant system R with the propositional constant t
(“the conjunction of all truths”) has the following axioms and rules
(Anderson & Belnap 1975, ch. V; ↔ is defined by A ↔
B = (A → B) & (B → A)):

Self-implication.

A → A

Prefixing.

(A → B) → ((C → A) → (C → B))

Contraction.

(A → (A → B)) → (A → B)

Permutation.

(A → (B → C)) → (B → (A → C))

& Elimination.

(A & B) → A, (A & B) → B

& Introduction.

((A → B) & (A → C)) → (A → (B &
C))

∨ Introduction.

A → (A
∨
B), B → (A
∨
B)

∨ Elimination.

((A → C) & (B → C)) → ((A
∨
B) → C)

Distribution.

(A & (B
∨
C)) → ((A &
B)
∨ C)

Double Negation.

¬¬A → A

Contraposition.

(A → ¬B) → (B → ¬A)

Ax. t.

A ↔ (t → A)

Modus Ponens.

A, A → B / B

Adjunction.

A, B / A & B

A relevant version RD of Mally's deontic system may
be defined as follows:

The language is the same as the language of R,
except that we write V instead of t, add the propositional constant U
and the unary connective !, and define Λ, ∩, f and ∞
as in Mally's system.

Axiomatization: add Mally's Axioms I-V to the axioms and rules
of R.

RD has the following properties.

Axioms I, II and III may be replaced by the following three
simpler
axioms:[8]

There are 12 mismatches between RD and Mally's
expectations: (5), (12)-(13), (15), (19)-(21), (23) and (24)-(26) are
not derivable even though Mally did not regard these formulas as
surprising, and (30) is a theorem even though Mally viewed it as
surprising.

Formulas (34) and (35) (the formulas which Mally viewed as the most
surprising theorems of his system) are in a sense stranger than
Menger's theorem A ↔ !A because the latter theorem is derivable in
RD supplemented with (34) or (35) while neither (34)
nor (35) is derivable in RD supplemented with A ↔
!A.[11]

Although most of Mally's surprising theorems are not derivable in
RD, this has nothing to do with Mally's own reasons
for regarding these theorems as surprising. They are not derivable in
RD because they depend on fallacies of relevance.
Mally never referred to such fallacies to explain his surprise. His
considerations were quite different, as we have already described.

RD is closely related to Anderson's relevant
deontic logic ARD, which is defined as
R supplemented with the following two axioms (Anderson
1967, 1968, McArthur 1981; Anderson used the unary connective O instead
of !):

ARD1( → ) is not a theorem of
RD+ARD2.[13]
This formula does not occur in Mally's
book. According to Anderson, Bohnert (1945) was the first one to
propose
it.[14]

ARD2 is not a theorem of
RD+ARD1.[15]
This
formula does not occur in Mally's book, but Mally endorsed the
corresponding informal principle: “a person who wills correctly does
not will (not even implicitly) the negation of what he wills; correct
willing is free of
contradictions.”[16]

RD supplemented with ARD1( → ) and ARD2 has
the same theorems as
ARD.[17]

Anderson's system has several problematical features (McArthur 1981,
Goble 1999, 2001) and RD shares most of these
features. But we will not go into this issue here. It is at any rate
clear that RD is better than Mally's original
system.

It was recently pointed out that it also possible to base Mally's deontic logic on intuitionistic propositional logic IPC rather than classical propositional logic (Lokhorst 2013; see also Centrone 2013).

Heyting's intuitionistic propositional calculus IPC has the
following axioms and rules (see Van Dalen 2002 and the entry on intuitionistic logic):

A → (B → A)

(A → (B → C)) → ((A → B) → (A →
C))

(A & B) → A

(A & B) → B

A → (B → (A & B))

A → (A ∨ B)

B → (A ∨ B)

(A → C) → ((B → C) → ((A ∨ B) →
C))

⊥ → A

modus ponens

substitution

If we add to these axioms and rules the following:

Abbreviations: ¬ A = A → ⊥, A ↔ B = (A →
B) & (B → A), T = A → A,

then we can formulate ID (an intuitionistic reformulation of Mally's deontic logic) as IPC plus Mally's axioms I – V and

The intuitionistic reformulation of Mally's deontic logic that
we have proposed is successful in so far as it avoids both Menger's
and Mally's own objections while preserving almost all the theorems
that Mally noticed himself. However, it is unacceptable as a system
of deontic logic in its own right. We mention only two reasons:

1. Theorem A → !A is intuitively invalid. No deontic
system, except Mally's, has this theorem.

2. It is unclear how permission is to be represented. If we use
the standard definition (P A = ¬ !¬ A), then P A ↔ !A
is a theorem, but P A and !A are not equivalent according to the
ordinary use of the words “permitted” and “obligatory.”

The relevantist reformulation of Mally's system which was discussed above
does not have these defects.

Although ID is unacceptable as a system of deontic logic,
it does make sense as a system of lax logic, as we will now
show. Lax logic is used in the areas of hardware verification in
digital circuits and access control in secure systems, where !
expresses a notion of correctness up to constraints. The term “lax”
was chosen “to indicate the looseness associated with the notion of
correctness up to constraints” (Fairtlough and Mendler 1997,
p. 3). There is a considerable literature on lax logic
(Goldblatt 2011).

Mally's deontic logic and lax logic arose from quite different
considerations. It is therefore remarkable that the intuitionistic
reformulation of Mally's logic that we have discussed is identical
with lax logic PLL* plus (34).

Instead of changing the non-deontic propositional basis of Mally's
system, one might also modify the specifically deontic axioms and
rules. This might of course be done in various ways, but the following
approach works well without departing too much from Mally's original
assumptions.[26]

First, regard f as primitive and replace Mally's definition of f in
terms of → and ! (Def. f, the very first specifically deontic
postulate in Mally's book) by the following definition of ! in terms of
V and f :

Def. !.

!A = V f A

Second, replace Axiom I, which may also be written as (B → C)
→ ((A f B) → (A f C)), with the following rule of
inference:

Rule f.

B → C / (A f B) → (A f C)

We may then derive:

1.

B → C / !B → !C

[ Def. !, R f ]

2.

(!A & !B) → !(A & B)

[ Def. !, Ax. II ]

3.

!V

[ 1, Ax. IV, PC ]

4.

¬!Λ

[ 1, Ax. III(←), Ax. V, PC (ex
falso) ]

The so-called standard system of deontic logic KD
is defined as PC supplemented with 1-4 (except that !
is usually written as O: see the entry on
deontic logic),
so the new system is at
least as strong as KD. It is not difficult to see that
it is in fact identical with KD supplemented with OU
(Mally's Axiom IV) and the following definition of f : A f B = O(A
→ B). In modern deontic logic, the notion of commitment
is sometimes defined in this way.

Although (34) and (35) are not derivable, adding them would by no
means lead to the theoremhood of A ↔ !A.

There were only 12 mismatches in the case of RD, so
the new system does less justice to Mally's deontic expectations than
RD did. But it agrees better with his general outlook
because it is still based on classical propositional logic, a system to
which Mally did not object (not that he had much choice in the
1920s).

Many of Mally's surprising theorems are derivable in
KD, but they have, as it were, lost their sting: those
theorems lead to surprising consequences when combined with Mally's
Axiom I and his definition of f , but they are completely harmless
without these postulates.

The standard system of deontic logic has several problematical
features. The fact that each provable statement is obligatory is often
regarded as counterintuitive, and there are many other well-known
“paradoxes.” The revised version of Mally's system shares these
problematical features. But we will not discuss these issues here. The
standard system is at any rate better than Mally's original
proposal.

Mally's deontic logic is unacceptable for the reasons stated by
Menger (1939). But it is not as bad as it may seem at first sight. Only
relatively minor modifications are needed to turn it into a more
acceptable system. One may either change the non-deontic basis to get either a
system that is similar to Anderson's system or a system that is identical with intuitionistic propositional logic with double negation
as an obligation-like operator, or apply two patches to
the deontic postulates to obtain a system similar to standard deontic
logic.

Some authors have refused to view Mally's deontic logic as a “real”
deontic system and say that they “mention it only by way of curiosity”
(Meyer and Wieringa 1993, p. 4). The above shows that this
judgment is too harsh. It is only a small step, not a giant leap, from
Mally's system to modern systems of deontic logic, so Mally's
pioneering effort deserves rehabilitation rather than contempt.

MaGIC 2.2.
MaGIC (Matrix Generator for Implication Connectives) is a program
which finds matrices for implication connectives for a wide range of
propositional logics. MaGIC was written by John Slaney of the
Automated Reasoning Group, The Research School of Information Sciences
and Engineering, The Australian National University. MaGIC was written
for Unix and can easily be compiled under Linux, FreeBSD,
Mac OS X and similar operating systems. There is also a
version for Windows (Cygwin).

Acknowledgments

The author is very grateful to Lou Goble, whose extensive comments on
two earlier drafts led to many substantial improvements. The author
would also like to thank Lou Goble and Edgar Morscher for making their
papers on Anderson's and Mally's deontic logic available to him,
Robert K. Meyer for helping him find the matrices used in
note 10, and John Slaney for providing him with the proof of II*
mentioned in note 8.

The SEP would like to congratulate the National Endowment for the Humanities on its 50th anniversary and express our indebtedness for the five generous grants it awarded our project from 1997 to 2007.
Readers who have benefited from the SEP are encouraged to examine the NEH’s anniversary page and, if inspired to do so, send a testimonial to neh50@neh.gov.